The closing of 42nd street, a very busy crosstown road in New York City,
during Earth Day in April 1990, was expected to cause a traffic nightmare.
Instead, as reported in The New York Times on December 25, 1990, the flow of
traffic actually improved.

In 2003, the Cheonggyencheon stream restoration project began in Seoul,
removing a six-lane highway. The project opened in 2005, and besides substantial environmental
benefits, a speeding up of traffic was observed around the city.

Likewise, planners have called for the closing of parts of Main street in
Boston and parts of the road connecting the Borough and the Farringdon
underground stations, in London.

If closing roads might help traffic flow, the negative effects of expanding
a road network can be observed as well. For instance, in the late 1960s the
city of Stuttgart decided to open a new street to alleviate the downtown
traffic. Instead, the traffic congestion worsened and the authorities ended
up closing the street, which improved the traffic.

Remaining pillars of the Cheonggyencheon highway, which was removed.

Stories like these abound and as you might suspect, some mathematics is
lurking behind them all. Indeed, in 1968, the mathematician Dietrich Braess,
working at the Institute for Numerical and Applied Mathematics in Münster, Germany,
proved that "an extension of a road network by an
additional road can cause a redistribution of the flow in such a way that
the travel time increases." In his work Braess
assumed that the drivers will act selfishly, each of them choosing a route
based on their own perceived benefit, with no regard for the benefit of other
drivers. It's an assumption that reflects the harsh
conditions of rush hour traffic rather well!

The phenomenon Braess observed, now called the Braess paradox, is not really a
paradox, but just unexpected behaviour showing that we are not very well
equipped to predict the outcomes of collective interactions.

The closing of 42nd street and the Cheonggyencheon stream restoration
project are just reverse examples of the Braess paradox, where the removal
of one or more roads improves the travel time along a road network.

Still a little skeptical about this Braess paradox? In the next section we
will analyse the mathematics of a very simple example.

The case of the superfast road

The road network shown below connects locations A and B.

Road network

At rush hour cars enter the network at A at a rate of 1500 cars per hour, and drivers choose one of two routes, route 1, crossing bridge , or route 2, crossing bridge .

We will write and for the number of cars, per hour, that arrive a B via route 1 and route 2, respectively.

The bridges, and , are bottlenecks that slow traffic. We will suppose that the travel time through both bridges is directly proportional to the number of cars per hour, or flow of cars. Specifically, we assume that the travel time is minutes for bridge and minutes for bridge . The rest of both routes consists of fairly large roads with a travel time of minutes each. We must say that although our assumptions are meaningful, their determination for a real network is a difficult example of mathematical modelling.

John Nash, March 2008.

We want to know the expected distribution of traffic, that is the number of cars, per hour, on each route. To do so, we imagine that each driver has gone through the network many times, as is the case for someone driving every day at rush hour, and has developed a particular strategy, that is perceived as minimising travel time. Under this assumption, the travel time must be the same for all the drivers, otherwise there would be an incentive for some of the drivers to change their strategy. This is what is called a steady-state, or Nash equilibrium, named after the mathematician and Nobel laureate John F. Nash. One of Nash’s many contributions, was to analyse so called non-cooperative games, of which rush hour traffic is a good example.

Let us observe that a Nash equilibrium is very different from, say, the
equilibrium of the coffee cup sitting on my desk. A Nash equilibrium is
dynamic, that is to say, to be maintained it needs to be fed, in our case,
by the cars that enter the network at A every hour. That everybody
achieves the same travel time means that at equilibrium no one is better
off, although we have assumed that each driver acts selfishly, trying to
minimise their own travel time with no regard for other drivers' interests. In
other words, whether they want it or not, each driver is influenced by the
collection of all the drivers' decisions.

Now, the travel time, in minutes, for each of the two routes is

Route 1:

Route 2:

At equilibrium we can write

Furthermore, the number of cars , and must add up to the incoming flow. So,

Solving these two simultaneous equations, we find that

So, the traffic distributes evenly between the two routes, with a travel time of 27.5 minutes.

We now assume that our road network is expanded with the addition of a super fast crossroad , for which the travel time is 7 minutes.

Expanded road network

Will this addition to the network decrease travel time? Let’s see.

Drivers can now choose between three routes, the two previous ones and a new route 3 that goes through bridge , onto road and through bridge . As before, is the flow of cars arriving at B via route 1 and the flow of cars leaving A viaroute 2. Moreover, is the flow of cars on road . Then the number of cars, per hour, going through bridge must be , while the number of cars, per hour, going through bridge must be . So, the travel time on each of the three routes will be

Route 1:

Route 2:

Route 3:

Again, we want to find the distribution of the traffic between these three routes.

As before, the traffic will have reached a steady-state, or a Nash equilibrium, when the travel time is the same for all the drivers. So, at equilibrium, we have

which gives us the two equations

Moreover, we have

From these three simultaneous equations we can find the three unknowns, , and and then the common travel time for all the drivers:

This new travel time of minutes represents a increase from the previous time!

What happened? The superfast new road has proven to be too tempting for too
many drivers, causing bad congestion and affecting the performance of the
whole network. The drivers do not have any incentive to switch to the other
routes, because they all have the same travel time. So, everybody gets
stuck. In other words, selfish behavior has eroded the efficiency of the
network, increasing travel time by 20%. Economists refer to this
phenomenon as "the price to pay for
anarchy". However, if the drivers agree to avoid route c
completely, the travel time will decrease. This option is the same as
adopting a cooperative strategy in which drivers split between the two
preexisting routes. Actually, some road networks do have controller systems
directing the traffic. In such networks the Braess paradox will not occur.
It is only observed when drivers choose their own best routes.

Braess is a tricky paradox

It is easy to accept that when the incoming flow is sufficiently small, the
Braess paradox will not occur. Actually, it has been observed that drivers,
acting selfishly, change from their original routes to the superfast road
with no increase in travel time.

On the other hand, one would think that a significant increase in the
incoming flow of cars will make things worse. However, this is not always
the case. In our example, scientists have conjectured that at a very high
demand, there will be a "wisdom of the
crowds" effect under which the new road would not be used.
It appears, indeed, that the individual decisions within a large enough
group of drivers may optimise the travel time for all. This conjecture was
proved in 2009 by the mathematician Anna Nagurney, a professor in the
Isenberg School of Management at the University of Massachusetts.

A way of experimenting with these ideas in our small example would be to use some algebra: replace the numbers with variables and see what relations between the variables imply that the
paradox will or will not occur. That is to say, whether the travel time in
the expanded network is larger than the travel time in the original network,
or not.

Before we exit ...

... a few more comments are in order.

The Braess paradox appears in many contexts. For instance, the article If we all go for the blonde, uses a blonde of allegedly
great alluring power to take the place of our superfast road, producing the
same effect, a congested field. But if we keep to networks, the paradox has
been observed with data traveling in a network of computers and with power
being delivered on the grid. Moreover, in 2012 an international team of
researchers proved, theoretically as well as experimentally, that the Braess
paradox may occur in systems of electrons.

Getting somewhere?

Our example of the expanded road system shows that at equilibrium, the
distribution of cars in the network does not need to be optimal. This brings
us to another interesting concept, formulated by the economist Vilfredo
Pareto (1848-1923). Pareto declared a distribution of resources to be
optimal if no individual can be made better off without making at least one
other individual worse off. Such a distribution is called Pareto optimal.

In our example, the roads in the network are the resources. Ignoring the new
road makes everybody better off, from the stand point of decreasing travel
time. Thus, the equilibrium in the expanded network is an example of a Nash
equilibrium which is not Pareto optimal.

Finally, let us observe that a distribution of resources described as Pareto
optimal does not need to be fair in the social sense of the word. For
instance, an allocation of resources in which I hog everything and you have
nothing is Pareto optimal because the only way to improve your lot is for
me to lose something. Efforts have been made by some economists, among them
Ravi Kanbur of Cornell University, to reformulate the notion of Pareto
optimality, adding a quantitative way of measuring fairness.

About the author

Josefina (Lolina) Alvarez was born in Spain. She earned a doctorate in mathematics from the University of Buenos Aires in Argentina, and is currently professor emeritus of mathematics at New Mexico State University, in the United States. Her long time interest is to communicate mathematics to general audiences. Lolina lives in Santa Fe, New Mexico with husband Larry and dog Lily. She walks every week many kilometres along the beautiful trails of Northern New Mexico.
For more on her work, visit this website.

Comments

Thanks for your very well written exposition on Braess paradox. I have learned a lot of new things by reading it.

I'm really curious about the mentioned conjecture on the "wisdom of the crowds" effect and wish to learn more about it. Could you tell me the title of the reference in which Anna Nagurney proved the mentioned conjecture?

The extra route the author added took those choosing it from one bottleneck to the other. Who would build such a road and who would choose it as their path? To me this entire analysis is the equivalent of treating traffic flow as pressure in a pipe. Although not a bad first approximation, the behavior of mindless fluid is not an accurate predictor of human behavior. For instance, if one had added a 100 mile long bypass to avoid 1 mile of road, one would slow the net flow of fluid as some of the mindless fluid would spend an inordinate amount of time on the bypass. But people are not mindless and virtually 100% would ignore the 100 mile bypass, as I believe they would avoid the new road taking one through both bottlenecks.

First, the humans were not modelled as a fluid here but as greedy, selfish beings. Assuming no one (else) takes the new road, I would go for it giving me a 7.5 + 7 + 7.5 = 21.5 minute travel time – faster than that of anybody else. Soon most would follow me causing the two bridges to get more congested. This would also slow down those who did not change their behaviour. Now everyone would have a 33 minute travel time.

And, there are several real world examples of this happening in real cities. One "too good" road can sometimes cause severe imbalance in the city road network, which will slow everybody, even those not using that road.

PS. Using this model, no one would take your 100 mile bypass. It would not affect the system at all.

I would suggest that this is only true when one adds an additional transaction route (with inherent/intrinsic response time and throughput characteristics) to a model that starts out with an exclusive OR choice of one of two routes, each with only one bottleneck, such that the participants now have a "choice" to take a route that now includes the heretofore impossible case of two bottlenecks.

Queuing systems behave non-linearly after all.

Now, if the "bypass" in the example model had been one that bypassed BOTH existing bottlenecks, it is easily see that even if the new route is moderately slower in it's throughput/response time than the existing two choices; the sum of the system in net, improves. And, of course, if the bypass is intrinsically faster (as used in the example), the system as a whole improves even further.

So the Premise was not only not proven, it was a bit disingenuously stated. In fact: if you want less traffic (i.e. congestion, etc.) you should build more, BETTER (intelligently routed) roads...

Thank you for this fascinating, thought-provoking article.
It has been illuminating to read the comments too.

I KNOW that mathematics has a massive, vital role to play if we hope to solve the problems of, & accompanying, transport & travel.

There are many aspects to this problem which were not mentioned, however; and I feel the article could have gained authority, & the readers benefitted, from widening the scope a little in order to find & show facts & figures, WITH possible/ probable causes, effects, facts, figures, problems, solutions, etc.

Everything which affects, &/ or is affected by, this urgent problem - which is changing our planet for the worse, exponentially - must be included in the formulae which WILL be written, informing how to best and most quickly solve the problems.
These formulae will rely on many variables, including, e.g, public transport - existence & accessibility; how, & whether to, make cycling & walking safer & more pleasant; change in pollution levels; social attitudes & norms; health; etc.
And the immense financial repercussions of sharply reducing car usage, ownership; road deaths & people maimed; loss of demand for oil, cars, etc. - On what will Economies rely, as oil ceases to be their mainstay?

All of these must be addressed: mathematicians are the people to do that most efficiently.